Fallacies in Mathematics

Fallacies in mathematics refer to errors (or mistaken reasoning) that can occur when dealing with mathematical concepts (or arguments). These fallacies often arise from misunderstandings, flawed assumptions, or incorrect logic, leading to incorrect or contradictory conclusions. Let’s explore some common fallacies in mathematics.

Division by Zero: One of the most well-known fallacies is dividing a number by zero. Mathematically, division by zero is undefined because it leads to contradictory results.

Ambiguous Definitions: Fallacies can also arise from imprecise or ambiguous definitions. When terms or concepts are not clearly defined, confusion and errors can occur.

Circular Reasoning: Circular reasoning occurs when an argument relies on its own conclusion or assumes what it aims to prove. In mathematics, this can be a subtle fallacy that hides the lack of proper justification.

False Counterexamples: Counterexamples are useful in mathematics to disprove statements or conjectures. However, using false counterexamples can lead to fallacious reasoning.

Avoiding fallacies in mathematics requires a combination of careful reasoning, precision in definitions, and adherence to established principles and rules.

1. Serdaniel says:

Here the error is in the application of the property: √(a×b) = √(a) × √(b), which is valid if a≥0 and b≥0. Therefore, the equality √[(-1)×(-1)] = √(-1) × √(-1) is an error.

2. Silvio says:

Number 4 is wrong since you cant divide by (a-b) because it’s zero if a=b

3. MS says:

2: Relies on incorrectly not specifying which root signs are valid, like 1 = sqrt(1^2) = sqrt(1) = -1

3: Same as 2

4: Divides by 0

5: Same as 4

6: When we say inf+x = x, the equality is referring to effects on multiplication and converging sequences/integration, not further addition

7: sin(x) is an oscillating function, so its inverse is infinitely multivariate unless the range is restricted, as is clear from the graph shape

8: Same as 7 but only 1 oscillation

9: The assumption of correlated inequality only applies if they are all positive integers, since with negatives the ratios are 1/-1 = -1/1

10: Constant +c wasn’t included in any of the indefinite integrals

11: logs of negative numbers are undefined